I've seen the shroud in documentaries and I can't see how it is claimed
to be Christ's with much conviction at all.

Read John 19:39-40:

"And there came also Nicodemus, which at the first came to Jesus by
night, and brought a mixture of myrrh and aloes, about an hundred pound
weight. Then took they the body of Jesus, and wound it in linen clothes
with the spices, as the manner of the Jews is to bury."

They wound the garment about him and didn't just lay one over him.
I've heard that the manner of burial back then was to wrap people
mummie-style with 2-3 inch wide strips of cloth. If this is so, it
would certainly knock out any claims of authenticity of the shroud.
jn